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Full text of "Treatise On Analysis Vol-Ii"

252       XIV   INTEGRATION IN LOCALLY COMPACT GROUPS

* + > f(st)dt

rxi^i==rcci^isdt=

Jo         t         Jo       st

which proves our assertion.

(1423) Let Gbea locally compact group with neutral element e, and let iibea
(left orriqht) Haar measure on G. Then G is discrete if and only if n({e}) > 0;
and G is compact if and only (/>*(G) < + oo (i.e., if and only if  is a bounded
measure (13.20)).

It is clear that if G is compact then p is bounded. If G is discrete, {e} is an
open neighborhood of e, and hence /*({*}) > 0 because the support of A* is the
whole of G Conversely, let V be a compact neighborhood of e. If /*({*}) > 0,
we have u({s}) = i*({e}) for all seG, because \L is invariant; hence the
number of points of V is finite and ^OO/MW)- Since G is Hausdorff it
follows that G is discrete.

Suppose now that /* is bounded and (say) left-invariant. Consider the set
(E of finite subsets [sl9 s2, ... ,sn] in G such that st V n Sj V = 0 whenever
i^j. We have

n;(V) = ju(^V u j2 V u - -  u sn V) ^ /i(G),

hence 72^XG)/MOO- Hence G contains a subset {^,...,^n} having the
largest possible number of elements. For each j e G it then follows that *V
must meet at least one of the st V, that is to say, s e st VV J. Hence G is the
union of the sets ^VV""1, which are compact (12.10.4), and so G is compact.

(14.2.4) Let G be a locally compact group, V an open subset of G, and ju a
nonzero positive measure on V, having the following property: ifUis an open
subset ofV and if seG is such that sU c V, then the image of the measure
liv, induced by fJionU (13.1.8), under the homeomorphism x\-+sx (13.1.6), is the
measure /*sU induced by n on s\J. Then there exists a unique left Haar measure a
on G which induces \i on V.

For each s e G, let p, be the image of \i under the homeomorphism
xl_> sx of V onto jV. The restriction of jus to V n ^V is the image of /v iv n v
under the restriction of xt-*sx to s'^V n V. By hypothesis, this image is
/*vnsv- BY translation it follows that for all s, tin G the measures jus and ^
have the same restriction to ^V n rV. By virtue of (13.1.9), there exists a
positive measure a on G which induces /*, on sV for all s e G. Clearly a is left-
invariant, and is therefore the unique left Haar measure on G which induces /i
onV.< b, containing the support of /. Hence for each interval [c, d] in